The limiting behavior of global minimizers in non-reflexive Orlicz-Sobolev spaces

Abstract

Let Ω \Omega be a smooth, bounded N N -dimensional domain. For each p > N p>N , let Φ p \Phi _{p} be an N-function satisfying p Φ p ( t ) ≤ t Φ p ′ ( t ) p\Phi _{p}(t)\leq t\Phi _{p}^{\prime }(t) for all t > 0 t>0 , and let I p I_{p} be the energy functional associated with the equation − Δ Φ p u = f ( u ) -\Delta _{\Phi _{p}}u=f(u) in the Orlicz-Sobolev space W 0 1 , Φ p ( Ω ) W_{0}^{1,\Phi _{p}}(\Omega ) . We prove that I p I_{p} admits at least one global, nonnegative minimizer u p u_{p} which, as p → ∞ p\rightarrow \infty , converges uniformly on Ω ¯ \overline {\Omega } to the distance function to the boundary ∂ Ω \partial \Omega .